Paralogistic {actuar} R Documentation

## The Paralogistic Distribution

### Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Paralogistic distribution with parameters `shape` and `scale`.

### Usage

```dparalogis(x, shape, rate = 1, scale = 1/rate, log = FALSE)
pparalogis(q, shape, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
qparalogis(p, shape, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
rparalogis(n, shape, rate = 1, scale = 1/rate)
mparalogis(order, shape, rate = 1, scale = 1/rate)
levparalogis(limit, shape, rate = 1, scale = 1/rate,
order = 1)
```

### Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `shape, scale` parameters. Must be strictly positive. `rate` an alternative way to specify the scale. `log, log.p` logical; if `TRUE`, probabilities/densities p are returned as log(p). `lower.tail` logical; if `TRUE` (default), probabilities are P[X <= x], otherwise, P[X > x]. `order` order of the moment. `limit` limit of the loss variable.

### Details

The paralogistic distribution with parameters `shape` = a and `scale` = s has density:

f(x) = a^2 (x/s)^a / (x [1 + (x/s)^a]^(a + 1))

for x > 0, a > 0 and b > 0.

The kth raw moment of the random variable X is E[X^k], -shape < k < shape^2.

The kth limited moment at some limit d is E[min(X, d)^k], k > -shape and shape - k/shape not a negative integer.

### Value

`dparalogis` gives the density, `pparalogis` gives the distribution function, `qparalogis` gives the quantile function, `rparalogis` generates random deviates, `mparalogis` gives the kth raw moment, and `levparalogis` gives the kth moment of the limited loss variable.

Invalid arguments will result in return value `NaN`, with a warning.

### Note

`levparalogis` computes the limited expected value using `betaint`.

See Kleiber and Kotz (2003) for alternative names and parametrizations.

The `"distributions"` package vignette provides the interrelations between the continuous size distributions in actuar and the complete formulas underlying the above functions.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

### References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

### Examples

```exp(dparalogis(2, 3, 4, log = TRUE))
p <- (1:10)/10
pparalogis(qparalogis(p, 2, 3), 2, 3)

## variance
mparalogis(2, 2, 3) - mparalogis(1, 2, 3)^2

## case with shape - order/shape > 0
levparalogis(10, 2, 3, order = 2)

## case with shape - order/shape < 0
levparalogis(10, 1.25, 3, order = 2)
```

[Package actuar version 3.1-4 Index]